Encoding method for the compression of a video sequence

ABSTRACT

The invention relates to an encoding method for the compression of a video sequence by means of a three-dimensional wavelet transform. This method is based on a hierarchical subband encoding process leading to transform coefficients that constitute a hierarchical pyramid. A spatio-temporal orientation tree, in which the roots are formed with the pixels of the approximation subband and the offspring of each of these pixels is formed with the pixels of the higher subbands, defines the spatio-temporal relationship inside said pyramid. The initial subband structure of the wavelet transform is, in the encoding process, preserved by scanning the subbands one after the other in an order that respects the parent-off-spring dependencies formed in the tree, and flags “off/on ” are added to each coefficient of the tree in view of a progressive transmission of the most significant bits of the coefficients. According to the invention, an additional, specific one bit flag is added to each subband for giving an information about the overall state of its coefficients, said additional information about the parent-offspring dependencies of each subband being then used to either process said subband if said flag has a first one of its two possible values or skip it if said flag has the second one said two possible values.

The invention relates to an encoding method for the compression of avideo sequence divided in groups of frames (GOFs) decomposed by means ofa three-dimensional (3D) wavelet transform leading to a given number ofsuccessive resolution levels which correspond to the decompositionlevels of said transform, said method being based on a hierarchicalsubband encoding process leading from the original set of pictureelements (pixels) of each GOF to transform coefficients constituting ahierarchical pyramid, a spatio-temporal orientation tree—in which theroots are formed with the pixels of the approximation subband resultingfrom the 3D wavelet transform and the offspring of each of these pixelsis formed with the pixels of the higher subbands corresponding to theimage volume defined by these root pixels—defining the spatio-temporalrelationship inside said hierarchical pyramid, the initial subbandstructure of the 3D wavelet transform being preserved by scanning thesubbands one after the other in an order that respects theparent-offspring dependencies formed in said spatio-temporal tree, andspecific one bit flags being added to each coefficient of thespatio-temporal tree in view of a progressive transmission of the mostsignificant bits of the coefficients, these flags being such that atleast one of them describes the state of a set of pixels and at leastanother one describes the state of a single pixel.

Video streaming over heterogeneous networks requires a high scalabilitycapability, i.e. that means that parts of a bitstream can be decodedwithout a complete decoding of the sequence and can be combined toreconstruct the initial video information at lower spatial or temporalresolutions (spatial scalability, temporal scalability) or with lowerquality (SNR or bitrate scalability). A convenient way to achieve allthese three types of scalability (scalable, temporal, SNR) is athree-dimensional (3D, or 2D+t) wavelet decomposition of the input videosequence, after a motion compensation of said sequence. The document WO01/84847 (PHFR000044) describes a fully scalable method of video codingaccording to which a temporal (resp. spatial) scalability is obtained byperforming a motion estimation at each temporal resolution level (resp.at the highest spatial resolution level). Hierarchical encoding of theresulting spatio-temporal trees is performed by means of a new encodingmodule based on the technique named Fully Scalable Zerotree (FSZ). Anoverview of this fully scalable coding method can also be found in “AFully Scalable 3D Subband Video Codec”, by V. Bottreau, M. Bénetière, B.Felts and B. Pesquet-Popescu, Proceedings of IEEE Signal ProcessingSociety, 2001 International Conference on Image Processing,Thessaloniki, Greece, Oct. 7-10, 2001, pp. 1017-1020.

This previous technique is inspired of the so-called Set Partitioning InHierarchical Trees algorithm (SPIHT), the principles of which must firstbe recalled. The original SPIHT algorithm, described for instance in “Anew, fast, and efficient image codec based on set partitioning inhierarchical trees”, A. Said and W. A. Pearlman, IEEE Transactions onCircuits and Systems for Video Technology, vol. 6, n^(o) 3, June 1996,pp. 243-250, and, for its extension to the 3D case, for instance in “Anembedded wavelet video coder using three-dimensional set partitioning inhierarchical trees (SPIHT)”, B. J. Kim and W. A. Pearlman, Proceedingsof Data Compression Conference, Mar. 25-27, 1997, Snowbird, Utah, USA,pp. 251-260, is based on a key concept: a partial sorting of thecoefficients according to a decreasing magnitude, and the prediction ofthe absence of significant information across scales of the waveletdecomposition by exploiting self-similarity inherent in natural images.This means that if a coefficient is insignificant at the lowest scale ofthe wavelet decomposition, the coefficients corresponding to the samearea at the other scales have a high probability to be insignificanttoo. Basically, the SPIHT is an iterative algorithm that consists incomparing a set of pixels corresponding to the same image area atdifferent resolutions with a value called “level of significance”, fromthe maximal significance level found in the spatio-temporaldecomposition tree down to 0. For a given level, or bitplane, two passesare carried out: the sorting pass, which looks for zero-trees orsub-trees and sorts insignificant and significant coefficients, and therefinement pass, which sends the precision bits of the significantcoefficients. The SPIHT algorithm examines the wavelet coefficients fromthe highest level of the decomposition to the lowest one. Thiscorresponds to first considering the coefficients corresponding toimportant details located in the smallest scale subbands, withincreasing resolution, and then examining the smallest coefficients,which correspond to finer details. This justifies the “hierarchical”designation of the algorithm: the bits are sent by decreasing importanceof the details they represent, and a progressive bitstream is thusformed.

A tree structure, called spatial (or spatio-temporal in the 3D case)orientation tree, defines the spatial (or spatio-temporal) relationshipinside the hierarchical pyramid of wavelet coefficients. The roots ofthe trees are formed with the pixels of the approximation subband at thelowest resolution (“root” subband), while the pixels of the highersubbands corresponding to the image area (to the image volume, in the 3Dcase) defined by the root pixel form the offspring of this pixel. In the3D version of the SPIHT algorithm, each pixel of any subband but theleaves has 8 offspring pixels, and each pixel has only one parent (withone exception for this rule: in the root case, one pixel out of 8 has nooffspring). The following notations describe the parent-offspringrelationship:

O(x,y,z): set of coordinates of the direct offspring of the node (x,y,z)

D(x,y,z): set of coordinates of all descendants of the node (x,y,z);

H(x,y,z): set of coordinates of all spatio-temporal orientation treeroots (nodes in the highest pyramid level: spatio-temporal approximationsubband)L(x,y,z)=D(x,y,z)−O(x,y,z)(and an illustration of these dependencies is given in thethree-dimensional case in FIG. 1, where the notations are the following:TF=temporal frame, TAS=temporal approximation subband, CFTS=coefficientsin the spatio-temporal approximation subbands (or root coefficients),TDS.LRL=temporal detail subband at the last resolution level of thedecomposition, and TDS.HR=temporal detail subband at higher resolution).

The SPIHT algorithm makes use of three lists : the LIS (list ofinsignificant sets), the LIP (list of insignificant pixels), and the LSP(list of significant pixels). In all these lists, each entry isidentified by a set of coordinates (x,y,z). In the LIP and LIS, (x,y,z)represents a unique coefficient, while in the LIS it represents a set ofcoefficients D(x,y,z) or L(x,y,z), which are sub-trees of thespatio-temporal tree. To differentiate between them, the LIS entry is oftype A if it represents D(x,y,z), and of type B if it representsL(x,y,z). During the first pass (sorting pass), all the pixels of theLIP are tested and those that become significant are moved to the listLSP. Similarly, the sets of the LIS that become significant are removedfrom the list LIS and split into subsets that are placed at the end ofthe LIS and will be each examined in turn. The LSP contains the list ofsignificant pixels to be “refined” the n^(th) bit of the coefficient issent if this one is significant with respect to the level n.

To improve the global compression rate of the video coding system, it isthen usually advised to add an arithmetic encoder to the zero-treeencoding module. In other approaches, most of the time, the hierarchicaland arithmetic coding modules are considered separately. To efficientlycombine them in a single coding system, some modifications have to beperformed on the original SPIHT algorithm. Although the use of listsLIS, LIP and LSP in SPIHT facilitates the classification task, theselists are an obstacle to a geographic organization of the coefficients.The in-depth search performed when scanning for zero-trees does notexploit the redundancy inside subbands and makes harder thedetermination of a relevant context for the arithmetic coding (thecontext is the information that may have some influence on the currentpixel and particularly the information related to neighboring pixels).The manipulation of the lists LIS, LIP, LSP conducted by a set oflogical conditions makes the order of pixel scanning hardly predictable.The pixels belonging to the same 3D offspring tree but coming fromdifferent spatio-temporal subbands are encoded and put one after theother in the lists, which has for effect to mix the pixels of foreignsubbands. Thus, the geographic interdependencies between pixels of thesame subband are lost. Moreover, since the spatio-temporal subbandsresult from temporal or spatial filtering, the frames of the sequenceare filtered along privileged axes that give the orientation of thedetails. This orientation dependency is also lost when the SPIHTalgorithm is applied, because the scanning does not respect thegeographic order.

Furthermore, the bits resulting from the examination of the lists LIS,LIP, LSP and the signs of the coefficients have quite differentstatistical properties. The relevant contexts for one list can betotally different from another. For example, as the LIP represents theset of insignificant pixels, it is considered that if a pixel issurrounded by insignificant pixels, it has great chance to beinsignificant too, but, for the LSP, it cannot be necessarily deducedthat the refinement bit of an examined pixel is one (resp. zero) if therefinement bits of its neighbors are ones (resp. zeros) at a certainlevel of significance.

By using the technique described in the document WO 01/84847 alreadycited, the initial subband structure of the 3D wavelet transform can bepreserved, and a marker, or flag, added to each coefficient indicates towhich list LIS, LIP or LSP this coefficient belongs. More precisely, inthe method considered in said patent application, the wholespatio-temporal tree is fully scanned for each new bitplane. At the endof the first bitplane, all the offspring dependencies of the 3D volumehave been evaluated (this first scanning is therefore quite critical andmust absolutely respect the calculation order of the offspringdependencies described in FIG. 2, where the notations are the following:SA=spatial axes (s), TA=temporal axis (t), R=roots, FC=first children,SC=second children, and TC=third children). According to said method,the subbands are scanned one after the other in an order that respectsthe parent-offspring relationships, and at least two different flags,and preferably four, are added to the coefficients of thespatio-temporal tree:

-   A) at least one, and preferably two of them describe the state of a    set (trees or subtrees):    -   DIRECT_SET_INSIG (or FS1) if D(x,y,z) is still insignificant;    -   INDIRECT_SET_INSIG (or FS2) if L(x,y,z) is still insignificant.-   B) at least another one, and preferably the two other ones describe    the state of a single pixel:    -   SIG (or FP3) if the current pixel is significant;    -   INSIG (or FP4) if it is not significant, or if its significance        is to be analyzed (put by default to the pixels that are not        included in a zero-tree).        The main steps of the method are:-   1. Initialization:    -   Put flag FP4 to all the coefficients of the lowest        spatio-temporal subband;    -   Put flag FS1 to 7 over 8 coefficients of the lowest        spatio-temporal subband.-   2. Calculate and output MSL (the maximum significance level found in    the spatio-temporal decomposition tree).

3. From n=MSL down to 0, do a full exploration of the spatio-temporaltree (two main approaches are possible, as described in the followingparagraph: spatially-driven resolution scalability, andtemporally-driven resolution scalability), with, for each coefficient(x,y,z) of the spatio-temporal tree, the following actions a) setsignificance :   1) if flag FS1 is “on”, then output = S_(n) (D(x,y,z)).if S_(n) (D(x,y,z)) = 1, then :     - for each (x′,y′,z′) ∈ O(x,y,z),put flag FP4 ;     - remove flag FS1 from (x,y,z) ;       - if L(i,j)≠Ø, then put flag FS2.   2) if flag FS2 is “on”, then output = S_(n)(L(x,y,z)). if S_(n) (L(x,y,z)) = 1, then :     - for each (x′,y′,z′) ∈O(x,y,z), put flag FS1 ;       - remove flag FS2 from (x,y,z). b) pixelsignificance :   (1) if flag FP3 is on, then output = the n^(th) bit of(x,y,z). 2) if flag FP4 is on, then output = S_(n) (x,y,z).   ifS_(n)(x,y,z) = 1, then :        put flag FP3 on ;        output sign(x,y,z) ;        remove flag FP4.The frames are filtered along privileged axes (spatial or temporal) thatgive the orientations of the details. These orientations can be bettertaken into account by scanning the subband along the same directions.Using the indicated method, there are then two main ways of exploringthe spatio-temporal volume of coefficients depending on the chosenprivileged orientation, which may be either the spatial or the temporalaxis. Consequently, two types of “multi-scalable” bitstreams may beobtained, a first one lead by the spatial resolution, and a second onelead by the temporal resolution:(A) Spatially-Driven Resolution Scalability:

For each bitplane, the tree scanning is spatially oriented, since inthis scheme the spatial resolutions are fully explored one after theother as shown in FIG. 3, all the temporal resolutions beingsuccessively scanned inside each spatial scale. In other words, thetemporal frequency is higher than the spatial one. In order to have thepossibility to skip some part of the bitstream, resolution flags areintroduced in the bitstream. The scanning strategy leads to a videobitstream organized as indicated in FIG. 4, where the lines s and tcorrespond respectively to spatial and temporal decomposition levels(SDL and TDL), the flags A are flags separating two bitplanes, and theflags B are flags separating two spatial decomposition levels.

(B) Temporally-Driven Resolution Scalability:

For each bitplane, the tree scanning is temporally oriented, since inthis scheme the temporal resolutions are fully explored one after theother as shown in FIG. 5, all the spatial resolutions being successivelyscanned inside each temporal scale. This scanning strategy leads to avideo bitstream organized as indicated in FIG. 6, to be compared withFIG. 4 (the flags B now separate two temporal decomposition levels). Inboth cases, the three types of scalability (temporal, spatialresolution, SNR) are obtained: the SNR scalability is still availablesince the spatio-temporal scanning is inserted in a bitplane iterativeloop, and the temporal and spatial scalability are provided respectivelywith t_(max) possible frame rates and s_(max) possible display sizes(t=1 to 4 and s=1 to 4 in the described examples), with t=1corresponding to the minimum frame rate_(min), and s=1 corresponding tothe minimum display size.

With this method, thanks to the fixed subband scanning (replacing thescanning of the lists) and the recognition of the flags, a coherentgeographic context is restituted for each model: the initial subbandstructure of the 3D wavelet transform is preserved, and the flag addedto each coefficient indicates to which list LIS, LIP or LSP thiscoefficient belongs. The hierarchical and logical organization of theSPIHT is preserved, and in the same time moving a coefficient from alist to another is “virtually” done by changing its flag, the order ofreading being now not dependent of the changes performed by the logic ofthe SPIHT algorithm. This method, which better exploits the neighboringinfluence on the current pixel than those which combine classical SPIHTalgorithm and entropy coding (and leads to a “natural” context directlyissued from the transformed image, in conformity with the bitplaneapproach, and not from the bits resulting from the original SPIHTalgorithm in the refinement passes), improves the compression rate andtherefore the coding efficiency, as the context is really related to thebit being encoded.

However, the exhaustive scanning of all the spatio-temporal treesubbands rapidly leads to the following drawback: even at low decodingbitrate, a high computation load is observed, which is contradictorywith the requirements of nowadays video applications.

It is therefore an object of the invention to propose an encoding methodavoiding this drawback.

To this end, the invention relates to an encoding method such as definedin the introductory part of the description and which is moreovercharacterized in that an additional, specific one bit flag is added toeach subband of the spatio-temporal tree for giving an information aboutthe overall state of its coefficients, said additional information aboutthe parent-offspring dependencies of each subband being then used forthe following decision:

-   -   each subband has to be processed when its additional flag has        the first of its two possible values, called “on”, at least one        of its coefficients having a coefficient flag “on”;    -   each subband has to be skipped when its additional flag has the        second one of the two possible values, called “off”, all its        coefficients flags being “off”.

The technical solution thus proposed allows for each spatiotemporalsubband to add, prior to any calculation, an information (such as amarker, or flag) concerning its parent-offspring dependencies, in such away that if a particular subband is found to be not related to any othersubband according to this flag, its encoding\decoding process isskipped, thus avoiding heavy and useless computations. It should benoted that the proposed invention does not result in any modification ofthe FSZ output bitstream and therefore does not lead to any qualitydegradation of the later reconstructed video.

The present invention will now be described with reference to theaccompanying drawings in which:

FIG. 1 gives examples of parent-offspring dependencies in the 3D case,in the spatio-temporal orientation tree;

FIG. 2 illustrates the hierarchy of the subbands in said spatio-temporaltree;

FIG. 3 shows a spatially-driven scanning of the spatio-temporal tree;

FIG. 4 depicts a bitstream organization made possible by the ordered 3DSPIHT;

FIG. 5 shows a temporally-driven scanning of the spatio-temporal tree,and

FIG. 6 depicts the structure of the bitstream obtained with saidscanning.

As seen above, in the FSZ technique, the whole spatio-temporal treeresulting from the wavelet decomposition is fully scanned bitplane (orsignificance level) by bitplane, all the parent-offspring dependencies(illustrated in FIG. 1) being established during the first bitplaneprocessing. This hierarchical relationship determines the subbandscanning order that is followed for all the remaining bitplanes (both atencoder and decoder side, no distinction being therefore made betweenencoder and decoder in the following since they both strictly respectthe same order). As explained in the document WO 01/84847above-mentioned, the main steps of the FSZ algorithm are the following:

(A) the initialization step, during which only the lowestspatio-temporal subband coefficients are characterized by flags enablingthe beginning of the scanning process, all the other subbandcoefficients being initialized to zero;

(B) the scanning step, during which a full exploration of thespatio-temporal tree is performed for each bitplane in an order thatstrictly respects the parent-offspring dependencies formed in saidspatio-temporal tree.

During this in-depth scanning, the state of the spatio-temporal subbandcoefficients is virtually changed by turning ON or OFF their descriptionflags. The scanning of the spatio-temporal tree is fully exhaustive:every subband is reviewed, without any a priori assumption about thestate of its coefficients, which means that for each subband, everycoefficient is analyzed. However, when examining said FSZ technique indetails, one may remark that in the particular case when none of thefour possible flags (FS1=DIREC_SET_INSIG for insignificant set of directoffspring, FS2=INDIRECT_SET_INSIG for insignificant set of indirectoffspring, FP3=SIG for significant pixel, FP4=INSIG for insignificantpixel) is ON (equivalent to zero), not only none information is outputin the bitstream, but also none coefficient state is changed. In otherwords, the processing of such a coefficient is useless since it does notbring any additional information. This computational load overhead isparticularly important when a subband contains only such coefficients.Moreover, this situation is very frequent for the first bitplanes sinceevery subband, except the lowest one, is initialized to zero.

According to the present invention, it is therefore proposed to add toeach subband a flag SCAN that gives an indication of the overall stateof its coefficients. When ON (that is to say at least one coefficient ofthe subband has a flag different from zero), this flag allows theprocessing of the subband. When OFF (that is to say all the coefficientflags are equal to zero), the subband is skipped, since it is known thatneither any bit will be output nor any flag will be changed. Consideringthe two main steps of the original FSZ method, it is proposed, accordingto the invention, to initialize the SCAN flag to ON for the lowestspatio-temporal subband (this root subband must be scanned in any case)and to OFF for all the other subbands. Starting from the root subbandcoefficients, the method will then update the flags of the offspringaccording to the rules defined in FSZ. The SCAN flag of the subbandsthat contain these offspring coefficients are then set to ON since theywill have to be analyzed during a further sorting pass (for lowerbitplanes).

In short, the present invention proposes to modify the FSZ method (asoriginally described in the above-mentioned document) in the followingsteps, the added parts being written in italics) 1. Initialization:   -Put flag FP4 to all the coefficients of the lowest spatiotemporalsubband ;   - Put flag FS1 to 7 over 8 coefficients of the lowestspatiotemporal subband ;   - Put flag SCAN to ON to the lowestspatio-temporal subband ;   - Put flag SCAN to OFF to all the otherspatio-temporal subbands. 2. Calculate and output MSL. 3. From n = MSLdown to 0, do a full exploration of the spatio-temporal tree, with, foreach subband : A) if flag SCAN is OFF, skip the subband and go directlyto the next subband in the spatio- temporal tree ; B) if flag SCAN isON, then for each coefficient (x,y,z) of the spatio-temporal tree, thefollowing actions are provided :   a) set significance:     1) if flagFS1 is ON, then output = S_(n)(D(x,y,z)).   if S_(n)(D(x,y,z)) == 1,then :       - for each (x′,y′,z′) ∈ O(x,y,z) , put flag FP4 ;       -remove flag FS1 from (x,y,z) ;         - if L(i,j) ≠ 0, then put flagFS2 ; -put flag SCAN-ON for each subband that contains each (x′,y′,z′) ∈O(x,y,z) respectively.     2) if flag FS2 is ON, then output = S_(n)(L(x,y,z)).   if S_(n) (L(x,y,z))==1, then :       - for each (x′,y′,z′)∈ O(x,y,z) , put flag FS1 ;       - remove flag FS2 from (x,y,z) ; - putflag SCAN ON for each subband that contains each (x′,y′,z′) ∈ O(x,y,z)respectively.   - b) pixel significance :     1) if flag FP3 is ON, thenoutput = the nth bit of (x,y,z).     2) if flag FP4 is ON, then output =S_(n)(x,y,z).   if S_(n) (x,y,z) == 1, then :         put flag FP3 ON ;        output sign (x,y,z) ;         remove flag FP4.The advantage of the implementation of the method according to theinvention is a very noticeable complexity reduction of the FSZ method,without any modification of the final output bitstream. The complexityreduction is all the more important given that encoding/decoding bitrateis low, where only the most important bitplanes are processed and manysubbands have not been yet connected to others by any parent-offspringdependencies, that is to say that many subbands still have their flagSCAN set to OFF and are therefore not analyzed, contrary to what wasdone in the original FSZ algorithm.

1. An encoding method for the compression of a video sequence divided ingroups of frames (GOFs) decomposed by means of a three-dimensional (3D)wavelet transform leading to a given number of successive resolutionlevels which correspond to the decomposition levels of said transform,said method being based on a hierarchical subband encoding processleading from the original set of picture elements (pixels) of each GOFto transform coefficients constituting a hierarchical pyramid, aspatio-temporal orientation tree—in which the roots are formed with thepixels of the approximation subband resulting from the 3D wavelettransform and the offspring of each of these pixels is formed with thepixels of the higher subbands corresponding to the image volume definedby these root pixels—defining the spatio-temporal relationship insidesaid hierarchical pyramid, the initial subband structure of the 3Dwavelet transform being preserved by scanning the subbands one after theother in an order that respects the parent-offspring dependencies formedin said spatio-temporal tree, and specific one bit flags being added toeach coefficient of the spatio-temporal tree in view of a progressivetransmission of the most significant bits of the coefficients, theseflags being such that at least one of them describes the state of a setof pixels and at least another one describes the state of a singlepixel, said encoding method being further characterized in that anadditional, specific one bit flag is added to each subband of thespatio-temporal tree for giving an information about the overall stateof its coefficients, said additional information about theparent-offspring dependencies of each subband being then used for thefollowing decision: each subband has to be processed when its additionalflag has the first of its two possible values, called “on”, at least oneof its coefficients having a coefficient flag “on”; each subband has tobe skipped when its additional flag has the second one of the twopossible values, called “off”, all its coefficients flags being “off”.2. An encoding method according to claim 1, in which two flags describethe state of a set of pixels and are, for each coefficient (x,y,z) ofsaid spatio-temporal tree, FS1 if D(x,y,z) is still insignificant andFS2 if L(x,y,z) is still insignificant where D(x,y,z) is the set ofcoordinates of all the descendants of the node (x,y,z) andL(x,y,z)=D(x,y,z)=O(x,y,z), with O(x,y,z) being the set of coordinatesof the direct offspring of the node (x,y,z) and two flags describe thestate of a single pixel and are FP3 if the current pixel is significantand FP4 if it is not significant or if its significance is to beanalyzed, said encoding method being further characterized in that,after an initialization step where the flag FP4 is put to all thecoefficients of the lowest spatio-temporal subband, the flag FS1 is putto 7 over 8 coefficients of said lowest spatio-temporal subband, theadditional flag is put to the first one (“on”) of its two values for thelowest spatio-temporal subband and to the second one (“off”) for all theother subbands, and the maximum significance level MSL is calculated,the exploration of the spatio-temporal tree, implemented according tosaid scanning order, includes the following steps: From the bitplanen=MSL down to 0, do a full exploration of the spatiotemporal tree,where, for each subband: A) if said additional flag has the second oneof its values, skip the subband and go directly to the next subband inthe spatiotemporal tree; B) if said additional flag has the first one ofits values, for each coefficient (x,y,z) of the spatio-temporal tree,the following actions are provided:   a) set significance:     1) ifflag FS1 is ON, then output = S_(n) (D(x,y,z)).   if S_(n) (D(x,y,z)) ==1, then:       - for each (x′,y′,z′) ∈ O(x,y,z) , put flag FP4;       -remove flag FS1 from (x,y,z) ;         - if L(i,j) ≠ 0, then put flagFS2; - put said additional flag to the first one of its values for eachsubband that contains each (x′,y′,z′) ∈ O(x,y,z) respectively.     2) ifflag FS2 is ON, then output = S_(n) (L(x,y,z)).   if S_(n) (L(x,y,z))==1, then:       - for each (x′,y′,z′) ∈ O(x,y,z) , put flag FS1 ;      - remove flag FS2 from (x,y,z) ; -put said additional flag to thefirst one of its values for each subband that contains each (x′,y′,z′) ∈O(x,y,z) respectively.   b) pixel significance:     1) if flag FP3 isON, then output = the nth bit of (x,y,z).     2) if flag FP4 is ON, thenoutput = S_(n)(x,y,z).   if S_(n) (x,y,z) == 1, then:       - put flagFP3 ON;       - output sign (x,y,z) ;       - remove flag FP4.